CN104749564B - Many quantile methods of estimation of sea clutter Weibull amplitude distribution parameters - Google Patents

Abstract

The invention discloses many quantile methods of estimation of sea clutter Weibull amplitude distribution parameters.Implementation step is：(1) cumulative distribution function of the model is drawn according to the probability density function of sea clutter Weibull Amplitude Distributed Models；(2) three sample accumulated probabilities p are chosen₁, p₂, p₃, three are obtained with regard to sea clutter Weibull Amplitude Distributed Models form parameter β and the quantile equation of scale parameter η according to quantile definition；(3) the estimated value present invention for obtaining estimated value and scale parameter that the estimated value of quantile sum and (4) obtain form parameter using three quantile equations in three estimated values and step 2 obtained using clutter amplitude data can reduce interference of the anomalous scattering unit to sample, parameter estimation performance is improved, can be used for the target detection under sea clutter background.

Description

Multi-quantile estimation method for sea clutter Weibull amplitude distribution parameters

Technical Field

The invention belongs to the technical field of signal processing, and particularly relates to a multi-quantile estimation method for sea clutter Weibull amplitude distribution parameters, which can be used for target detection under a sea clutter background.

Background

The sea clutter is formed by mutually overlapping backscattering vectors of a large number of mutually independent sea surface scatterers and is a key factor influencing sea surface target detection and target motion parameter estimation. The optimal target detection method under the sea clutter background depends on model parameters of a sea clutter amplitude distribution model, the sea clutter amplitude distribution model changes along with the change of radar resolution and sea conditions, and the key of the sea surface target detection problem is how to effectively estimate the model parameters of the sea clutter amplitude distribution model. Under the condition of a low-resolution radar, the complex Gaussian model can well simulate the distribution of the sea clutter, and the amplitude of the sea clutter generally obeys Rayleigh distribution. When the high-resolution radar works under the condition of a small ground wiping angle, the amplitude distribution of the sea clutter has long trailing compared with Rayleigh distribution, and the sea clutter presents stronger non-Gaussian property. The Weibull distribution and lognormal distribution models are classical statistical models describing non-rayleigh envelope clutter. The Weibull distribution is a clutter model between Rayleigh distribution and lognormal distribution, and parameters of the Weibull distribution are properly adjusted to be changed into Rayleigh distribution or lognormal distribution. Therefore, the Weibull distribution model can accurately describe the amplitude distribution of the actual sea clutter in a wider range than the rayleigh distribution model and the log-normal distribution model.

In adaptive detection, the threshold value is typically related to a parameter of the amplitude distribution model. Therefore, in order to substantially keep the constant false alarm rate constant and obtain a higher detection probability under a certain condition, the estimation of the amplitude distribution model parameters is particularly important. Maximum likelihood estimation and moment estimation are common methods for Weibull distribution parameter estimation. Maximum likelihood Estimation of Weibull Distribution model parameters is discussed in the literature "Estimation of Three-parameter Weibull Distribution" (see Hongzhu Qiao, ChrisP. Tsokos. "Estimation of the Three-parameter Weibull Distribution [ J ]". Mathesics and Computer Distribution, 1995,39: 173-. The literature "estimation of Weibull distribution parameters in the case of full samples" (see, Cibotian, estimation of Weibull distribution parameters in the case of full samples [ J ] ", application of probability statistics, 2003, 19 (3): 259-266) discusses a rectangular estimation of Weibull distribution parameters using logarithmic samples. Weibull distribution maximum likelihood estimation needs to search and solve transcendental equations in a real number range, and the solving time is long. The Weibull distribution moment estimation relates to a Gamma function, a display expression cannot be obtained, and the calculation is complex. And in practical application scenes, a plurality of abnormal scattering units are included, and great errors can be generated when the parameters of the clutter amplitude distribution model are estimated by using the methods.

Disclosure of Invention

The invention aims to provide a multi-quantile estimation method for sea clutter Weibull amplitude distribution model parameters, which aims to improve the estimation accuracy of the shape parameters and the scale parameters of a sea clutter amplitude distribution model under the condition that an abnormal scattering unit exists.

In order to achieve the technical purpose, the technical scheme of the invention comprises the following steps:

(1) obtaining an expression of an accumulative distribution function F (x, eta, beta) according to a probability density function F (x, eta, beta) of a sea clutter Weibull amplitude distribution model, wherein eta represents a scale parameter of the amplitude distribution model, beta represents a shape parameter of the amplitude distribution model, and x represents sea clutter amplitude;

(2) selecting three samples to accumulate the probability p₁＝0.5，p₂＝0.75，p₃0.5, according to its corresponding quantile Respectively satisfy:the following three quantile equations are obtained:

(3) the radar transmitter transmits pulse signals, and the radar receiver receives echo data X formed by sea surface scattering_rFrom X_rSelecting a distance unit only containing clutter data, and selecting N clutter amplitude data in total: x is the number of₁,x₂,....,x_NAnd arranging the sequences in ascending order to obtain an increasing sequence x₍₁₎,x₍₂₎,....,x_(N)And calculating an estimated value of each quantile by using the incremental sequence:

wherein,to representIs determined by the estimated value of (c),to representIs determined by the estimated value of (c),to representEstimated value of (Np)₁Indicating the position of the sample at the first quantile, Np₂Indicating the position of the sample at the second quantile, Np₃Indicates the position of the sample at the third quantile, round (Np)₁) Indicates the closest Np₁Integer of (1), round (Np)₂) Indicates the closest Np₂Integer of (1), round (Np)₃) Indicates the closest Np₃An integer of (d);

(4) using three estimated values obtainedAndseparately substituting in the three quantile equations of step (2)Andsolving the equation to obtain the estimated value of the shape parameter of the sea clutter Weibull amplitude distribution modelAnd an estimate of a scale parameter

Compared with the prior art, the invention has the following advantages:

1) because only the samples with the left side of the quantile point are considered, the invention can effectively avoid the influence of the abnormal scattering unit with larger amplitude on the right side of the quantile point, greatly improve the performance of parameter estimation and obtain the steady estimation of the shape parameter and the scale parameter.

2) Because the shape parameters of the sea clutter Weibull amplitude distribution model are explicitly expressed by utilizing the ratio of the two branch points, compared with a moment estimation method and a maximum likelihood estimation method, the method has the advantages of simpler calculation and short solving time;

drawings

FIG. 1 is a flow chart of an implementation of the present invention;

FIG. 2 is a graph of the probability density function result of a sea clutter Weibull amplitude distribution model obtained by the present invention and a prior art method;

FIG. 3 is a graph of a probability density function of changing the conventional coordinate system of FIG. 2 to a logarithmic coordinate system;

FIG. 4 is a diagram illustrating a cumulative distribution function of a sea clutter Weibull amplitude distribution model obtained by the present invention and a conventional method.

Detailed Description

The invention will be further described with reference to the accompanying drawings in which:

referring to fig. 1, the implementation steps of the invention are as follows:

step 1, calculating an expression of an accumulative distribution function F (x, eta, beta) according to a probability density function F (x, eta, beta) of a sea clutter Weibull amplitude distribution model.

Integrating the probability density function F (x, eta, beta) of the sea clutter Weibull amplitude distribution model to obtain an expression of an accumulative distribution function F (x, eta, beta) of the sea clutter Weibull amplitude distribution model:

wherein,

in the formula, x represents the sea clutter amplitude, eta represents a scale parameter and represents the median of Weibull amplitude distribution, beta represents a shape parameter and represents the skewness of the Weibull amplitude distribution, and the value range of beta is more than 0 and less than or equal to 2.

Step 2, selecting three sample cumulative probabilities p₁，p₂And p is₃And defining three quantile point equations according to the quantile points.

(2.1) defining the localization point with the sea clutter amplitude x corresponding to the sample accumulation probability p as x_pThe quantile x_pThe condition satisfied is as follows<2>Shown in the figure:

p＝p(x≤x_p)＝F(x_p,η,β)<2>

(2.2) selecting three sample cumulative probabilities p₁＝0.5，p₂＝0.75，p₃0.5, according to its corresponding quantileRespectively satisfy: the following three quantile equations are obtained:

and 3, solving the estimation value of each quantile.

(3.1) receiving echo data X formed by sea surface scattering by a radar receiver_rFrom X_rN clutter amplitude data of the first 6 distance units only containing the clutter are selected, and the N clutter amplitude data are rearranged into a row x₁,x₂,....,x_N；

(3.2) selecting clutter amplitude data x₁,x₂,....,x_NArranged in ascending order to obtain an increasing sequence x₍₁₎,x₍₂₎,....,x_(N)And calculating an estimated value of each quantile by using the incremental sequence:

wherein,to representIs determined by the estimated value of (c),to representIs determined by the estimated value of (c),to representEstimated value of (Np)₁Indicates the position of the first quantile of the sample, Np₂Indicates the position of the second quantile of the sample, Np₃Indicates the position of the third quantile in the sample, round (Np)₁) Indicates the closest Np₁Integer of (1), round (Np)₂) Indicates the closest Np₂Integer of (1), round (Np)₃) Indicates the closest Np₃Is an integer of (1).

Step 4, using the three estimated valuesAnd step 2 of the formula<3>The three expressed quantile equations obtain the estimated value of the shape parameter of the sea clutter Weibull amplitude distribution modelAnd an estimate of a scale parameter

(4.1) utilization of the formula in step 3<4>Evaluation of three quantilesAndrespectively substitute the formula in step 2<3>In the equation of three quantilesAndthree new quantile equations are obtained as follows:

(4.2) use formula<5>The middle three new quantile equations to obtain the estimated value of each quantileAndrelationship between shape parameter β and scale parameter η:

(4.3) use formula<6>Estimate of the first point of attachmentAnd an estimate of the second split pointIs obtained by the expression ofAndthe ratio of (A) to (B) is:

from the formula <7>, it can be seen that the ratio of any two sub-sites is independent of the scale parameter η, and thus an estimated value of the shape parameter is obtained

(4.4) general formula<8>Estimation of a represented shape parameterSubstituted type<6>Estimate of the third quantile ofIn the expression of (a), an estimate of the scale parameter η is obtained:

based on the steps 1 to 4, multi-quantile estimation of sea clutter Weibull amplitude distribution model parameters is achieved.

The effect of the present invention will be further explained with the simulation experiment.

1. Simulation parameters

The sea clutter data used in the simulation experiments were actually measured sea clutter data recorded by an IPIX radar at the university of McMaster, canada, 19931109_191449. mat. The radar has the transmitting frequency of 9.39GHz, the antenna height of 30m, the antenna gain of 45.7dB, the polarization modes of HH and VV, the echo data of each range unit comprises 131072 sampling points, and each group of data has 14 range units. The simulation data used in the experiment is the first 6 groups of data without targets in VV polarization in actually measured sea clutter data (19931109-191449. mat) recorded by an IPIX radar, and pure clutter data.

2. Content of simulation experiment

In a simulation experiment, the method and the moment estimation method are respectively adopted to obtain the estimation of sea clutter Weibull amplitude distribution model parameters, and the effects of the two estimation methods are analyzed and compared through a mean square error detection (MSD) method and a modified mean square error detection (MMSD) method.

Simulation experiment 1

Firstly, N pure clutter data of the first 6 distance units are selected from an actually measured sea clutter data matrix, wherein N is 131072 × 6, and an estimation value of a shape parameter obtained by a moment estimation method in a simulation experiment isThe estimated value of the scale parameter isIn the method of the present invention, the estimated values of the shape parameters obtained through the above 4 steps areThe estimated value of the scale parameter is

Simulation experiment 2

On the basis that the simulation experiment 1 respectively adopts the method and the moment estimation method to obtain the shape parameter and the scale parameter estimation value, an empirical probability density curve of actually measured sea clutter data is drawn, and then a probability density curve of a sea clutter Weibull amplitude distribution model obtained by the method and the moment estimation method is drawn respectively, as shown in FIG. 2, the horizontal axis of the graph 2 represents sea clutter amplitude, and the vertical axis represents probability density.

As can be seen from fig. 2, in the part with a smaller amplitude at the front end, the fitting result of the method of the present invention to the sea clutter is closer to the real clutter distribution.

Simulation experiment 3

The probability density function of the sea clutter has a long tailing characteristic, and the fitting degree of each probability density curve of a tailing part to the clutter is difficult to observe under a constant coordinate system, so that the result of the simulation experiment 2 is drawn under a logarithmic coordinate system, as shown in fig. 3, the horizontal axis of the graph 3 represents the amplitude of the sea clutter, and the vertical axis represents the probability density.

It can be seen from figure 3 that the fitting of the present invention to sea clutter is still better than the moment estimator in the "tail" part.

Simulation experiment 4

Drawing an empirical cumulative distribution function curve of the actually measured sea clutter data, and then respectively drawing a cumulative distribution function curve of a sea clutter Weibull amplitude distribution model obtained by adopting the method and the moment estimation method, as shown in FIG. 4, wherein the horizontal axis of the graph 4 represents the sea clutter amplitude, and the vertical axis represents the cumulative distribution function of the sea clutter Weibull amplitude distribution model.

It can be seen from fig. 4 that the fitting effect of the present invention to the sea clutter accumulation distribution function is better.

Simulation experiment 5

A fitting test method is adopted to quantify the experimental comparison result of the method and the moment estimation method, mean square error is firstly used for testing MSD, the result of which estimation method is closer to the amplitude model of the real sea clutter is integrally tested, then modified mean square error test MMSD is introduced, and the result of which estimation method is better in the tailing part of the sea clutter is tested. MSD of the method is 0.0016 and MSD of a moment estimation method is 0.0022 through simulation calculation; the MMSD of the invention is 1.8816e-005, the MMSD of the moment estimation method is 3.9906e-005,

the smaller the values of MSD and MMSD, the closer the corresponding sea clutter Weibull amplitude distribution model is to the real clutter amplitude model. It is clear that the estimation of the present invention works better both globally and "tailing".

In conclusion, the multi-quantile estimation method for the sea clutter Weibull amplitude distribution parameters can reduce the interference of abnormal scattering units with larger amplitudes on samples, improve the estimation performance and simply and effectively estimate the sea clutter Weibull amplitude distribution model parameters.

Claims (3)

1. A multi-quantile estimation method for sea clutter Weibull amplitude distribution parameters is characterized by comprising the following steps:

(1) obtaining an expression of an accumulative distribution function F (x, eta, beta) according to a probability density function F (x, eta, beta) of a sea clutter Weibull amplitude distribution model, wherein eta represents a scale parameter of the amplitude distribution model, beta represents a shape parameter of the model, and x represents the sea clutter amplitude;

(2) selecting three samples to accumulate the probability p₁＝0.5，p₂＝0.75，p₃0.5, according to its corresponding quantileRespectively satisfy:the following three quantile equations are obtained:

$1-\exp \&lsqb;-{\left(\frac{x_{p_1}}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\&rsqb;=p_1$

$1-\exp \&lsqb;-{\left(\frac{x_{p_2}}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\&rsqb;=p_2$

$1-\exp \&lsqb;-{\left(\frac{x_{p_3}}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\&rsqb;=p_3;$

(3) the radar transmitter transmits pulse signals, and the radar receiver receives echo data X formed by sea surface scattering_rFrom X_rSelecting a distance unit only containing clutter data, and selecting N clutter amplitude data in total: x is the number of₁,x₂,....,x_NAnd arranging the sequences in ascending order to obtain an increasing sequence x₍₁₎,x₍₂₎,....,x_(N)And calculating an estimated value of each quantile by using the incremental sequence:

$\left\{\begin{array}{c}{\hat{x}}_{p_1}=x_{\left(n_1\right)},n_1=round\left(N{p}_1\right)\\ {\hat{x}}_{p_2}=x_{\left(n_2\right)},n_2=round\left(N{p}_2\right)\\ {\hat{x}}_{p_3}=x_{\left(n_3\right)},n_3=round\left(N{p}_3\right)\end{array}\right.$

wherein,to representIs determined by the estimated value of (c),to representIs determined by the estimated value of (c),to representEstimated value of (Np)₁Indicating the position of the sample at the first quantile, Np₂Indicating the position of the sample at the second quantile, Np₃Indicates the position of the sample at the third quantile, round (Np)₁) Indicates the closest Np₁Integer of (1), round (Np)₂) Indicates the closest Np₂Integer of (1), round (Np)₃) Indicates the closest Np₃An integer of (d);

(4) using three estimated values obtainedAndseparately substituting in the three quantile equations of step (2)Andsolving the equation to obtain the estimated value of the shape parameter of the sea clutter Weibull amplitude distribution modelAnd an estimate of a scale parameter

2. The method for estimating the multi-quantile position of the sea clutter Weibull amplitude distribution parameter according to claim 1, wherein the expression of the cumulative distribution function F (x, η, β) is obtained by integrating the expression F (x, η, β) of the probability density function of the sea clutter Weibull amplitude distribution model in step (1), that is:

$F(x,\mathrm{\&eta;},\mathrm{\&beta;})={\&Integral;}_0^{x}f(x,\mathrm{\&eta;},\mathrm{\&beta;})dx=1-\exp \&lsqb;-{\left(\frac{x}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\&rsqb;,x\&GreaterEqual;0$

wherein,

$f(x,\mathrm{\&eta;},\mathrm{\&beta;})=\frac{\mathrm{\&beta;}}{\mathrm{\&eta;}}{\left(\frac{x}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}-1}\exp \&lsqb;-{\left(\frac{x}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\&rsqb;,x\&GreaterEqual;0,$

in the formula, x represents the sea clutter amplitude, eta represents a scale parameter and represents the median of Weibull amplitude distribution, beta represents a shape parameter and represents the skewness of the Weibull amplitude distribution, and the value range of beta is 0< beta < 2.

3. The multi-quantile estimation method of sea clutter Weibull amplitude distribution parameters of claim 1, wherein the step (4) of solving the quantile equation obtains the estimated value of the shape parameters of the sea clutter Weibull amplitude distribution modelAnd an estimate of a scale parameterThe method comprises the following steps:

4a) using the estimated values of the three quantiles obtained in step (3)Andseparately substituting in the three quantile equations of step (2)Andthree new quantile equations are obtained as follows:

$1-\exp \&lsqb;-{\left(\frac{{\hat{x}}_{p_1}}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\&rsqb;=p_1$

$1-\exp \&lsqb;-{\left(\frac{{\hat{x}}_{p_2}}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\&rsqb;=p_2$

$1-\exp \&lsqb;-{\left(\frac{{\hat{x}}_{p_3}}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\&rsqb;=p_3;$

4b) obtaining the estimated value of each quantile by using the three new quantile equations in the step 4a)Andrelationship between shape parameter β and scale parameter η:

${\hat{x}}_{p_1}=\mathrm{\&eta;}\sqrt[\mathrm{\&beta;}]{-ln(1-p_1)},{\hat{x}}_{p_2}=\mathrm{\&eta;}\sqrt[\mathrm{\&beta;}]{-ln(1-p_2)},{\hat{x}}_{p_3}=\mathrm{\&eta;}\sqrt[\mathrm{\&beta;}]{-ln(1-p_3)};$

4c) using the estimated value of the first point of attachment obtained in step 4b)And an estimate of the second split pointIs obtained by the expression ofAndthe ratio of (A) to (B) is:

$\frac{{\hat{x}}_{p_1}}{{\hat{x}}_{p_2}}=\frac{\mathrm{\&eta;}\sqrt[\mathrm{\&beta;}]{-ln(1-p_1)}}{\mathrm{\&eta;}\sqrt[\mathrm{\&beta;}]{-ln(1-p_2)}}=\sqrt[\mathrm{\&beta;}]{\frac{ln(1-p_1)}{ln(1-p_2)}};$

that is, the ratio of any two sub-points is independent of the scale parameter η, and the estimated value of the shape parameter is obtained

$\widehat{\mathrm{\&beta;}}={\log }_{\frac{{\hat{x}}_{p_{{}_2}}}{{\hat{x}}_{p_1}}}\frac{\ln (1-p_2)}{\ln (1-p_1)}=\ln \left(\frac{\ln (1-p_2)}{\ln (1-p_1)}\right)/h({\hat{x}}_{p_2}/{\hat{x}}_{p_1});$

4d) The estimated value of the shape parameter obtained in the step 4c) is usedSubstituting into the estimate of the third quantile in step 4b)In the expression of (a), an estimate of the scale parameter η is obtained: